Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 4] Infinitely Small and Large Quantities 35
293. For x *0 determine the orders of smallness relative to
x of the functions:
*\
*)
b)
^*
d) 1 cos *'*
\ +x e) tan A: sin A:.
c) $/*'-
294. Prove that the length of an infinitesimal arc of a circle
of constant radius is equivalent to the length of its chord.
295. Can we say that an infinitesimally small segment and
an infinitesimally small semicircle constructed on this segment
as a diameter are equivalent?
Using the theorem of the ratio of two infinitesimals, find
296. lim
si" 3*' s 5*
!"
. 298. . lim^
arc sin _^= 299. lim
297. lim
x ^o ln(l--*)
1 -*
, f
~
300. Prove that when x *0 the quantities ~ and Y\ +xl
are equivalent. Using this result, demonstrate that
small we have the approximate equality
when \x\ is
VT+T1 + . (1)
Applying formula (1), approximate the following:
a) 1/L06; b) 1/0^7; c) /lO; d) /T20
and compare
the values obtained with tabular data.
301. Prove that when x we have the following approximate
equalities accurate to terms of order x 2
:
b)
c) (1 +x) n &\ + nx (n is a positive integer);
d) log(l+x) = Afx,
where Af = log e = 0.43429...
Using these formulas, approximate:
*> 02 ; 2 > 0^7 ; 3 >
I